Answer :

The all possible rational zeros of f(x)=x⁶-2x⁴+7x²+25 are ±1, ±5 and ±25.

To find the possible rational zeros of the polynomial function f(x)=x⁶-2x⁴+7x²+25, we can use the Rational Root Theorem.

  • The Rational Root Theorem states that any rational solution of a polynomial equation, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient.

  • For the polynomial f(x), the constant term is 25, and the leading coefficient (coefficient of x⁶) is 1. Therefore, the factors of the constant term 25 are ±1, ±5, ±25, and the factors of the leading coefficient 1 are ±1.

  • Thus, the possible rational zeros are: ±1, ±5 and ±25.
    This gives us a total of six possible rational zeros: +1, -1, +5, -5, +25, -25.

However, it's important to note that these are just the possible rational zeros. These values should be tested in the function to determine if they are actual zeros.